Consider the wave equation

Question

Problems:

Sound Waves

Consider the wave equation

                  ∂²s/∂t² = c2Δs

when the solution s admits spherical symmetry, ie, s(t,x,y,z)=v(t,r), where r = √x² + y² + z², the wave equation becomes:

              ∂²v/∂t² = c²/r² ∂/∂r(r² ∂v/∂r) (1)

making the substitution

                v(r,t) = h(t,r)/r

for some twice differentiable function h, show that (1) becomes

               ∂²h/at² = c² ∂²h/∂r²

hence, show that the general solution reads

          v(r,t) - 1/r (ƒ(r-ct) +g(r+ct))

for any twice differentiable functions f and g.